CN105204332B - Dead band and sluggish compound Sandwich system method for estimating state are contained based on Non-smooth surface observer - Google Patents
Dead band and sluggish compound Sandwich system method for estimating state are contained based on Non-smooth surface observer Download PDFInfo
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Abstract
The invention discloses a kind of compound Sandwich system method for estimating state for containing dead band and sluggishness based on Non-smooth surface observer, first with key item separation principle and switching function, from simple to complex, use for reference the band dead band built, band gap and with sluggish Sandwich system Non-smooth surface state space equation, build the Non-smooth surface state space equation that energy accurate description contains dead band and sluggish compound Sandwich system；Secondly according to the compound Non-smooth surface sandwich state space equation of structure, when system meets the existence consition of observer, the Non-smooth surface state estimation observer that construction can change and automatically switch with compound Non-smooth surface Sandwich system operation interval.This method have the advantage that：Such system is more accurately described by introducing switching function；Disclosed compound Non-smooth surface observer can automatically switch with the switching between system work area；Compared with traditional observer, using this method observer can more accurately estimating system state value.
Description
Technical field
The invention belongs to the state estimation field of nonlinear system, more particularly to a kind of containing based on Non-smooth surface observer
Dead band and sluggish compound Sandwich system method for estimating state.
Background technology
For any control system, the Assignment of Closed-Loop Poles of system is either realized, the decoupling of system is still realized
All it be unable to do without feedback of status.Meanwhile, the state of accurate estimating system be also realize system optimal control and fault diagnosis it is important
Foundation.However, the state of system is not necessarily all easy to directly detect, some state variables can not even be detected at all
Arrive.Therefore, researcher solves problem above often through structural regime observer.But, for particular system, if not root
State estimation is carried out according to system actual conditions construction observer, the inaccurate of state estimation will be caused.Estimate according to inaccurate
Meter state can not realize the accurate control to system, in fault diagnosis, and inaccurate state estimation also results in the mistake of failure
Report and fail to report.Therefore, the state estimation observer of particular system state can accurately be estimated and can realize that exact failure is pre- by building
The failure prediction observer of report is particularly important.
Dead band, gap and sluggishness are most common nonsmooth nonlinearities characteristics.Dead band is widely present in direct current generator, machinery
In transmission system, Hydraulic Power Transmission System, electro-mechanical system and power amplifier system.Gap is widely present in gear mechanism biography
In dynamic system, motor-driven valve, digital circuit, sensor and hydraulic system.In recent years, using the intelligence of memorial alloy, piezoelectric ceramics
Performs device is widely used in precision finishing machine, boat due to having the advantages that positioning precision is high, driving force is big and it is fast to respond
In the alignment system of the precision equipments such as the flexible mechanical arm and astronomical telescope of its aircraft, but memorial alloy and piezoelectric ceramics
In there are sluggish this nonsmooth nonlinearities, therefore, sluggishness is widely present in the flexible machine of precision finishing machine, space shuttle
In the alignment system of the precision equipment such as tool arm and astronomical telescope.In addition, the radio circuit and lock phase commonly used in communication system
Loop also includes sluggish Sandwich system.And in engineering in practice, dead band, gap and Hysteresis Nonlinear characteristic are frequently not lonely
Vertical, but be connected with other traditional links, it is clipped between two linear dynamic links, is referred to as Non-smooth surface sandwich system
System.Non-smooth surface Sandwich system with dead band, gap and sluggishness can be unified to be described with the structured flowchart shown in Fig. 1.
But, in actual engineering, many systems are not typical Non-smooth surface Sandwich system as shown in Figure 1, example
Such as, a flexible mechanical arm is made up of the executing agency of amplifying circuit, direct current generator, piezoelectric ceramics and robotic arm, equally
Amplifying circuit is considered as a linear element L1, and direct current generator has dead band and regards a dead band link DZ as, and piezoelectric ceramics is deposited
A sluggish link HS is considered as in lagging characteristics, finally, the executing agency of robotic arm can be regarded as a linear loop
Save L2.Therefore, whole system can be expressed as the structure shown in Fig. 2, and its Non-smooth surface link connects one late by a dead band
Stagnant composition.
Therefore, invention defines the new Non-smooth surface Sandwich system of a class：If the centre of Non-smooth surface Sandwich system
Link more than one, but multiple Non-smooth surface links are cascaded, such Non-smooth surface Sandwich system is referred to as compound non-light
Sliding Sandwich system, the system shown in Fig. 2 is exactly containing dead band and sluggish compound Sandwich system.
The study hotspot that corresponding observer always is control engineering field is constructed for particular system.From D.J.
Since Luenberger (1971) proposes famous Luenberger observers in 1970's, for Time-Invariant Linear System
The design courses of observer are just ripe.But it is different for nonlinear system, first, the controllability of nonlinear system
It is a local characteristicses；Second, the controllability of linear system is unrelated with system input, is solely dependent upon the structure of system in itself, and
The controllability of nonlinear system is not only relevant with system architecture, while also relevant with system input.Just because of nonlinear system
Complexity, for nonlinear system, we are difficult to find a unified observer building method, often for a certain class
Nonlinear system constructs the specific observer of a class.
In terms of special system Design of Observer, Ma Kemao, Ma Ping (2003) is to meeting the non-linear of Lipschitz conditions
System has carried out Design of Observer research.Zhu Fang comes (2004), and reduced dimension observer enters to class Lyapunov Nonlinear System Designs
Row state estimation.Zhou Shaowu, Zhang Jing (2001) propose a kind of design side of improved Lipschitz nonlinear systems observer
Method.Luan little Li etc. (2008) constructs the observer of time-delay systems with nonlinear uncertainties, and such system is entered using the observer
Row robust control.Han Chunyan etc. (2009) builds specific observer for Stochastic Delay Systems and carries out state estimation.Wu Min etc.
(2008) state estimator design containing inconsiderable transition occurrence diagram is completed.A.Lj.Juloski. (2002,2003) are waited to be directed to
Piecewise-linear system designs a kind of Luenberger types switching observer and carries out state estimation.A.Alessandri etc. (2001) profits
The observer gain matrix of piecewise linearity switching system is solved with shared Lyapunov functional based methods.Zhou Zupeng, Tan Yonghong
(2011,2012), which devise Non-smooth surface observer to single band dead band, gap and sluggish Sandwich system and carried out state, to be estimated
Meter.
But, above method for estimating state carries out Design of Observer both for particular system, all false in these methods
If system is to be entirely capable of seeing or given only one of which Non-smooth surface link, and compound Non-smooth surface Sandwich system is in certain workspace
Between it is endless all can see and with two series connection Non-smooth surface links.Therefore, state is being carried out to compound Non-smooth surface Sandwich system
When estimating Design of Observer, it is necessary to while the new problem after considering that system is endless and all can seeing and be combined.Therefore, the present invention will be opened up
Complication system State Observer Theory is opened up, the accurate status containing dead band and sluggish compound Sandwich system state variable is solved
Estimation problem, the present invention has certain scientific theory meaning and actual application prospect.
The content of the invention
For the deficiency of above-mentioned technology, contain dead band and slow the invention discloses a kind of being directed to based on Non-smooth surface observer
The method for estimating state of stagnant compound Sandwich system, with band dead band, band gap and with sluggish single Non-smooth surface sandwich system
System compares, and can solve the single insurmountable Complex Problem of Non-smooth surface observer using this method, be compared with traditional observer,
Using this method observer can more accurately estimating system state value.
Realizing the technical scheme of the object of the invention is：
The compound Sandwich system method for estimating state containing dead band and sluggishness based on Non-smooth surface observer, comprising as follows
Step：
Step 1：Using key item separation principle and switching function, from simple to complex, the band dead band built, band are used for reference
Gap and with sluggish Sandwich system Non-smooth surface state space equation, build can accurate description contain dead band and sluggish compound three
The Non-smooth surface state space equation of Mingzhi's system；
Step 2：The compound Non-smooth surface sandwich state space equation built according to step 1, when system meets observer
During existence consition, the Non-smooth surface state that construction can change and automatically switch with compound Non-smooth surface Sandwich system operation interval is estimated
Observer is counted, and provides the existence condition and convergent displacement of corresponding Non-smooth surface state estimation observer.
The step 1 comprises the following steps：
(1) the leading portion linear element L containing dead band and sluggish compound Sandwich system1(), as shown in formula (1)：
Rear end linear element L containing dead band and sluggish compound Sandwich system2(), as shown in formula (2)：
(2) dead band and sluggish modeling
Mathematical modeling between dead band link DZ input and output, as shown in formula (3)：
The mathematical modeling of sluggish input/output relation, as shown in formula (4)：
Wherein,
Wherein,yi∈R1×1,u∈R1×1, vi∈R1×1, i=
1,2, x1iAnd x2iI-th of state variable of leading portion linear element and rear end linear element is represented respectively；It is state
Transfer matrix,It is input matrix, yi∈R1×1It is output variable, niRepresent the dimension of i-th of linear subsystem, u ∈
R1×1It is input variable, v1∈R1×1It is sluggish input variable while being also the output variable in dead band, v2∈R1×1It is the defeated of sluggishness
Go out variable.zi(k) be i-th of gap output, wiIt is i-th of gap weight shared in sluggishness is constituted, Bi() is single
The input-output function in individual gap, n is used for building the number in the gap of sluggishness；m1bIt is the slope in gap, m1dIt is the oblique of dead band
Rate；D1bIt is the width in gap, D1dIt is the width in dead band.
The step 2 comprises the following steps：
(1) the Non-smooth surface observer of the system is constructed, according to formula (1), (2), (3) and (4), construction contains dead band and sluggishness
Compound Sandwich system state observer it is as follows：
Wherein,
Wherein,It is feedback matrix；
(2) existence condition and convergent displacement of fixed pattern (5) observer are given:
Initialization system meets following condition：
The state variable x of condition 1 is bounded, i.e.,‖x(k)‖m≤xb, xb≥0；Wherein ‖ ‖mRepresent m norms, xbRepresent
The maximum boundary value of state variable；
The initial error of the observer of condition 2 is bounded, i.e. ‖ e (1) ‖m≤eb, eb≥0；ebRepresent observer initial error
Maximum boundary value；
The shift-matrix A of the leading portion linear subsystem of condition 31Characteristic value all in unit circle；
Theorem：The compound Sandwich system containing dead band and sluggishness for meeting three above assumed condition, can be with structure
Make the observer as shown in formula (5) and state estimation is carried out to the system, if the feedback matrix K of selection observer2So that (A2-
K2C22) characteristic value in unit circle, then the evaluated error of the observer shown in formula (5) eventually converges to zero, its
In, A2Represent the transfer matrix of rear end linear system, K2Represent the feedback matrix of rear end linear system, C22Represent that rear end is linearly
The output matrix of system.
The method have the advantages that：
1. taken into full account dead band and sluggish non-linear and Non-smooth surface characteristic and two kinds of series connection composite attribute, by
In compound Sandwich system, introduce switching function and more accurately describe such system, it is established by compound function formula (4)
Accurate mathematical modeling is to describe such system, and formula (4) links together dead band and sluggish Non-smooth surface characteristic, solve as
What accurate description is combined the problem of sandwich.
2. on the basis of the accurate mathematical model of the system of structure, the compound Non-smooth surface state observer of construction estimates its shape
State value, wherein compound Non-smooth surface observer can automatically switch with the switching between system work area, is particularly wrapped in formula (5)
Contain the compound relationship between expression of two Non-smooth surface links of connection, successfully solve compound Non-smooth surface Sandwich system state observation
The structure problem of device, and give the existence condition and convergent displacement of such State Observer.
3. with single band dead band, band gap, the Non-smooth surface sandwich Non-smooth surface observer of band sluggishness compares, construction is answered
Close the state estimation problem that Non-smooth surface Sandwich system successfully solves the series connection of multiple Non-smooth surface links, this be in the past method without
What method was solved.
4. compared with conventional linear observer, the compound Non-smooth surface method for estimating state of construction can more accurately estimating system
State value.
Brief description of the drawings
The structure of Fig. 1 Non-smooth surface Sandwich systems；
Fig. 2 contains the structure of dead band and sluggish compound Sandwich system；
Fig. 3 dead bands link DZ input/output relation figure；
Input-output characteristic figure sluggish Fig. 4；
The input/output relation figure in i-th of gap of Fig. 5；
The state estimation result figure of Fig. 6 observers of the present invention；
The state estimation result figure of Fig. 7 conventional linear observers；
The state estimation application condition figure of Fig. 8 observers of the present invention and conventional linear observer.
Embodiment
Present invention is further elaborated with reference to embodiment and accompanying drawing, but is not limitation of the invention：
Embodiment
The compound Sandwich system method for estimating state containing dead band and sluggishness based on Non-smooth surface observer, comprising as follows
Step：
Step 1：Using key item separation principle and switching function, from simple to complex, the band dead band built, band are used for reference
Gap and with sluggish Sandwich system Non-smooth surface state space equation, build can accurate description contain dead band and sluggish compound three
The Non-smooth surface state space equation of Mingzhi's system.
Comprise the following steps in step 1：
(1) the leading portion linear element L containing dead band and sluggish compound Sandwich system1：
Rear end linear element L containing dead band and sluggish compound Sandwich system2：
(2) dead band and sluggish modeling
Dead band link DZ input and output relation are shown in Fig. 3, and the mathematical modeling between its input and output is：
Wherein, m1d=m2d=1, D1d=D2d=0.01.
Sluggish input-output characteristic is as shown in figure 4, the mathematical modeling of its input/output relation is：
HS：Sluggishness is formed by stacking by n=7 gap, and the parameter in 7 gaps is as follows：
wi=1, m1b=m2b=1, D1b=D2b=ci/ 2, (i=1,2 ..., 7),
c1=0.14, c2=0.12, c3=0.1, c4=0.08, c5=0.06, c6=0.04, c7=0.02, wherein ciRepresent
The width in i-th of gap, the input/output relation in i-th of gap is as shown in Figure 5.
Step 2：The compound Non-smooth surface sandwich state space equation built according to step 1, when system meets observer
During existence consition, the Non-smooth surface state that construction can change and automatically switch with compound Non-smooth surface Sandwich system operation interval is estimated
Observer is counted, and provides the existence condition and convergent displacement of corresponding Non-smooth surface state estimation observer.
Step 2 comprises the following steps：
(1) the Non-smooth surface state observer as shown in formula (5) proposed in the content of the invention is built as follows：
Wherein,
c1=0.14, c2=0.12, c3=0.1, c4=0.08, c5=0.06, c6=0.04, c7=0.02,
Wherein, K2=[0.1 0.1]T∈R2×1It is feedback matrix.
(2) convergence of analysis Non-smooth surface observer formula (6)：
Embodiment meets three conditions of convergence theorem：
Condition 1：The state variable x of embodiment is bounded, i.e.,‖x(k)‖1≤ 100, wherein ‖ ‖1Represent 1 norm；
Condition 2：The initial error of observer is bounded in embodiment, i.e. ‖ e (1) ‖1≤20；
Condition 3：The shift-matrix A of the leading portion linear subsystem of embodiment1Characteristic value be [0.8,0.45]TAll in unit
In circle；
In embodiment, K is chosen2=[0.1 0.1]T, then (A2-K2C2) characteristic value for [0.8000+0.0316i,
0.8000-0.0316i]TIn unit circle, it can be seen from the convergence theorem in step 2, the state estimation of system is most at last
Converge to actual value.
In order to illustrate the method for estimating state of the present invention and the validity of Non-smooth surface state observer, the present embodiment is also constructed
A kind of traditional Systems with Linear Observation device carries out state estimation to same system, shown in Systems with Linear Observation device such as formula (7)：
Wherein,
Wherein, A represents transfer matrix, and B represents input matrix, KlGain matrix is represented, the gain of Systems with Linear Observation device is set
Matrix Kl=[0 0 0.1 0.1]T, and the initial state value of given observer isEmulation system
The initial value of system is x (0)=[0,0,0,0]T, the sampling period is 0.01 second, and simulation time is 3 seconds.
As seen from Figure 6, method for estimating state of the invention can tracking system well actual condition value, obtain accurate
State estimation.As seen from Figure 7, traditional Systems with Linear Observation device can not accurate estimating system x21And x (k)22(k) two states
Value.As seen from Figure 8, for x21And x (k)22(k) two states, the evaluated error of traditional Systems with Linear Observation device method of estimation is remote
More than the evaluated error of the method for estimating state of the present invention.
Claims (1)
1. the compound Sandwich system method for estimating state containing dead band and sluggishness based on Non-smooth surface observer, using pass
Key separation principle and switching function, from simple to complex, use for reference the band dead band built, band gap and with sluggish sandwich system
System Non-smooth surface state space equation, builds the Non-smooth surface state that energy accurate description contains dead band and sluggish compound Sandwich system
The step of space equation, it is characterised in that：Also include
According to the compound Non-smooth surface sandwich state space equation of structure, when system meets the existence consition of observer, structure
The Non-smooth surface state estimation observer that can change with compound Non-smooth surface Sandwich system operation interval and automatically switch is made, and is provided
It is the step of existence condition and convergent displacement of corresponding Non-smooth surface state estimation observer, specific as follows：
（1）The Non-smooth surface observer of the system is constructed, the state observation containing dead band and sluggish compound Sandwich system is constructed
Device is as follows：
Wherein,It is feedback matrix；
（2）To fixed pattern（5）The existence condition and convergent displacement of observer:
Initialization system meets following condition：
The state variable of condition 1It is bounded, i.e.,WhereinRepresent
M norms,Represent the maximum boundary value of state variable；
The initial error of the observer of condition 2 is bounded, i.e.,Represent observer
The maximum boundary value of initial error；
The transfer matrix of the leading portion linear subsystem of condition 3Characteristic value all in unit circle；
Theorem：The compound Sandwich system containing dead band and sluggishness for meeting three above assumed condition, can be constructed such as
Formula（5）Shown observer carries out state estimation to the system, if the feedback matrix of selection observerSo thatCharacteristic value in unit circle, then formula（5）The evaluated error of shown observer eventually restrains
To zero, wherein,The transfer matrix of rear end linear system is represented,The feedback matrix of rear end linear system is represented,Represent
The output matrix of rear end linear system.
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| CN108279569B (en) * | 2018-01-12 | 2020-08-21 | 桂林电子科技大学 | State estimation method for hysteresis sandwich system with fault |
| CN112731809B (en) * | 2020-12-21 | 2023-02-28 | 桂林电子科技大学 | State and fault estimation method for dead zone sandwich system |
| CN112987683B (en) * | 2021-01-08 | 2022-02-18 | 桂林电子科技大学 | Fault positioning method for dead zone non-smooth sandwich system |
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